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Tutorial:Quality upcycling math

2025-02-23T21:46:22Z

Rorshan: fixed broken tabber tabs

{{Languages}}
How do we get the most amount of [[Quality|legendary]] items out of an upcycling plant?

The answer is not quite as straight forward as we'd like it to be, because it depends on a number of factors, luckily there is a finite number of possibilities of what the modules can be, and for the sake of simplicity this tutorial will ignore the productivity gain from [[Technologies#Space_Age|infinite technologies]]. But first, a presentation of the results.

== Best ratios ==
The table below shows the best ratio for [[Quality module 3|quality]] to [[Productivity module 3|productivity modules]] in the crafting machines, while the [[Recycler|recyclers]] always take only quality modules. The values are not given in whole numbers because often it is not just a single crafting machine per tier that will be used, then the ratios can change between different crafting machines in the same tier. e.g. "3.67 quality / 1.33 productivity" could have 4 machines where 3 have a ratio 4 to 1, and one a ratio 3 to 2.

<tabber>
common modules =
The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.
{| class="wikitable"
! Crafting machine || Modules for common items || Modules for uncommon items || Modules for rare items || Modules for epic || Modules for legendary || Percentage yield || Items recycled*
|-
| Chemical plant || 3 quality 0 productivity || 3 quality 0 productivity || 3 quality 0 productivity || 3 quality 0 productivity || 0 quality 3 productivity || 0.034014% || 2940
|-
| Assembling machine 3 || 4 quality 0 productivity || 4 quality 0 productivity || 4 quality 0 productivity || 4 quality 0 productivity || 0 quality 4 productivity || 0.046275% || 2161
|-
| Foundry || 4 quality 0 productivity || 4 quality 0 productivity || 4 quality 0 productivity || 4 quality 0 productivity || 0 quality 4 productivity || 0.133814% || 747
|-
| Electromagnetic plant || 5 quality 0 productivity || 5 quality 0 productivity || 5 quality 0 productivity || 5 quality 0 productivity || 0 quality 5 productivity || 0.176712% || 566
|-
| Cryogenic plant || 6 quality 2 productivity || 6 quality 2 productivity || 6 quality 2 productivity || 6.5 quality 1.5 productivity || 0 quality 8 productivity || 0.119134%|| 840
|}
<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
|-|
uncommon modules =
The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.
{| class="wikitable"
! Crafting machine || Modules for common items || Modules for uncommon items || Modules for rare items || Modules for epic || Modules for legendary || Percentage yield || Items recycled*
|-
| Chemical plant || 3 quality 0 productivity || 3 quality 0 productivity || 3 quality 0 productivity || 3 quality 0 productivity || 0 quality 3 productivity || 0.059498% || 1681
|-
| Assembling machine 3 || 3.75 quality 0.25 productivity || 3.75 quality 0.25 productivity || 3.8 quality 0.2 productivity || 3.9 quality 0.1 productivity || 0 quality 4 productivity || 0.082296% || 1216
|-
| Foundry || 4 quality 0 productivity || 4 quality 0 productivity || 4 quality 0 productivity || 4 quality 0 productivity || 0 quality 4 productivity || 0.243699% || 410
|-
| Electromagnetic plant || 4.7 quality 0.3 productivity || 4.67 quality 0.33 productivity || 4.75 quality 0.25 productivity || 4.9 quality 0.1 productivity || 0 quality 5 productivity || 0.324189% || 309
|-
| Cryogenic plant || 4.6 quality 3.4 productivity || 4.6 quality 3.4 productivity || 4.67 quality 3.33 productivity || 5 quality 3 productivity || 0 quality 8 productivity || 0.257621% || 389
|}
<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
|-|
rare modules =
The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.
{| class="wikitable"
! Crafting machine || Modules for common items || Modules for uncommon items || Modules for rare items || Modules for epic || Modules for legendary || Percentage yield || Items recycled*
|-
| Chemical plant || 2.8 quality 0.2 productivity || 2.8 quality 0.2 productivity || 2.9 quality 0.1 productivity || 2.9 quality 0.1 productivity || 0 quality 3 productivity || 0.100660% || 994
|-
| Assembling machine 3 || 3 quality 1 productivity || 3.1 quality 0.9 productivity || 3.2 quality 0.8 productivity || 3.33 quality 0.67 productivity || 0 quality 4 productivity || 0.145220% || 689
|-
| Foundry || 3.5 quality 0.5 productivity || 3.5 quality 0.5 productivity || 3.6 quality 0.4 productivity || 3.9 quality 0.1 productivity || 0 quality 4 productivity || 0.424039% || 236
|-
| Electromagnetic plant || 3.6 quality 1.4 productivity || 3.6 quality 1.4 productivity || 3.6 quality 1.4 productivity || 3.9 quality 1.1 productivity || 0 quality 5 productivity || 0.588510% || 170
|-
| Cryogenic plant || 3.6 quality 4.4 productivity || 3.6 quality 4.4 productivity || 3.6 quality 4.4 productivity || 3.9 quality 4.1 productivity || 0 quality 8 productivity || 0.565030% || 177
|}
<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
|-|
epic modules =
The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.
{| class="wikitable"
! Crafting machine || Modules for common items || Modules for uncommon items || Modules for rare items || Modules for epic || Modules for legendary || Percentage yield || Items recycled*
|-
| Chemical plant || 2.33 quality 0.67 productivity || 2.4 quality 0.6 productivity || 2.4 quality 0.6 productivity || 2.4 quality 0.6 productivity || 0 quality 3 productivity || 0.152486% || 656
|-
| Assembling machine 3 || 2.5 quality 1.5 productivity || 2.5 quality 1.5 productivity || 2.6 quality 1.4 productivity || 2.8 quality 1.2 productivity || 0 quality 4 productivity || 0.232966% || 430
|-
| Foundry || 2.7 quality 1.3 productivity || 2.7 quality 1.3 productivity || 2.75 quality 1.25 productivity || 3 quality 1 productivity || 0 quality 4 productivity || 0.664130% || 151
|-
| Electromagnetic plant || 2.6 quality 2.4 productivity || 2.6 quality 2.4 productivity || 2.67 quality 2.33 productivity || 2.9 quality 2.1 productivity || 0 quality 5 productivity || 0.974700% || 103
|-
| Cryogenic plant || 2.6 quality 5.4 productivity || 2.6 quality 5.4 productivity || 2.6 quality 4.4 productivity || 2.8 quality 4.2 productivity || 0 quality 8 productivity || 1.122444% || 90
|}
<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
|-|
legendary modules=
The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.
{| class="wikitable"
! Crafting machine || Modules for common items || Modules for uncommon items || Modules for rare items || Modules for epic || Modules for legendary || Percentage yield || Items recycled*
|-
| Chemical plant || 1.67 quality 1.33 productivity || 1.67 quality 1.33 productivity || 1.67 quality 1.333 productivity || 1.8 quality 1.2 productivity || 0 quality 3 productivity || 0.344061% || 291
|-
| Assembling machine 3 || 1.67 quality 2.33 productivity || 1.67 quality 2.33 productivity || 1.67 quality 2.33 productivity || 1.8 quality 2.2 productivity || 0 quality 4 productivity || 0.586191% || 171
|-
| Foundry || 1.4 quality 2.6 productivity || 1.4 quality 2.6 productivity || 1.4 quality 2.6 productivity || 1.5 quality 2.5 productivity || 0 quality 4 productivity || 1.624266% || 62
|-
| Electromagnetic plant || 1 quality 4 productivity || 1 quality 4 productivity || 1 quality 4 productivity || 1 quality 4 productivity || 0 quality 5 productivity || 2.722332% || 37
|-
| Cryogenic plant || 0 quality 8 productivity || 0 quality 8 productivity || 0 quality 8 productivity || 0 quality 8 productivity || 0 quality 8 productivity || 4.835199% || 21
|}
<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
</tabber>

=== Number of crafting machines ===
If we assume a constant input stream of uncommon items, which will always fill back up, we can additionally figure out what ratio of items will be inside the system at once, and with that we can figure out how many crafting machines we need per tier of quality. This is done by setting <math>m_{1,k+1} = 100 \, \% - m_{2,k+1} - m_{3,k+1} - m_{4,k+1}</math> after each iteration and further adjusting for the crafting machines change in speed dependant on the modules (assuming the machines which only house productivity modules are not additionally boosted by [[Speed module 3|speed moduled]] [[beacon]]s). See the calculations further below for a full explanation of the calculations.

<tabber>
per recycler=
{| class="wikitable"
! Crafting machine || Recyclers || Machines producing common items || Machines producing uncommon items || Machines producing rare items || Machines producing epic items || Machines producing legendary items
|-
| Chemical plant || 1 || 4.418130519 || 0.037216657 || 1.294369664 || 0.009779 || 0.431955
|-
| Assembling machine 3 || 1 || 4.733868899 || 0.052316655 || 1.527502978 || 0.015336 || 0.52576
|-
| Foundry || 1 || 4.314446 || 0.082198 || 1.683088 || 0.03016 || 0.58448
|-
| Electromagnetic plant || 1 || 4.494435 || 0.116985 || 1.93479 || 0.049665 || 0.60795
|-
| Cryogenic plant || 1 || 5.23556 || 0.21384 || 2.53374 || 0.1034 || 0.594
|}
|-|
per legendary crafter (exact) =
{| class="wikitable"
! Crafting machine || Recyclers || Machines producing common items || Machines producing uncommon items || Machines producing rare items || Machines producing epic items || Machines producing legendary items
|-
| Chemical plant || 90.1223 || 331.8198 || 38.5325 || 11.2893 || 2.9653 || 1
|-
| Assembling machine 3 || 52.8432 || 208.4632 || 30.3992 || 9.8094 || 2.8783 || 1
|-
| Foundry || 24.7668 || 89.0483 || 19.9206 || 7.7712 || 2.8592 || 1
|-
| Electromagnetic plant || 16.5379 || 61.9418 || 17.4010 || 7.4908 || 3.1822 || 1
|-
| Cryogenic plant || 10.8718 || 47.4350 || 18.2124 || 8.8138 || 4.2654 || 1
|}
|-|
per legendary crafter (conservative)=
{| class="wikitable"
! Crafting machine || Recyclers || Machines producing common items || Machines producing uncommon items || Machines producing rare items || Machines producing epic items || Machines producing legendary items
|-
| Chemical plant || 53 || 198 || 23 || 7 || 2 || 1
|-
| Assembling machine 3 || 31 || 123 || 18 || 6 || 2 || 1
|-
| Foundry || 14 || 53 || 12 || 5 || 2 || 1
|-
| Electromagnetic plant || 14 || 56 || 16 || 7 || 3 || 1
|-
| Cryogenic plant || 9 || 41 || 16 || 8 || 4 || 1
|}
</tabber>

== The crafting machines ==
{| class="wikitable"
! Crafting machine || Module slots <math>N</math> || Base productivity bonus <math>p_0</math>
|-
| [[Chemical plant]] || 3 || +0%
|-
| [[Assembling machine 3]] || 4 || +0%
|-
| [[Foundry]] || 4 || +50%
|-
| [[Electromagnetic plant]] || 5 || +50%
|-
| [[Cryogenic plant]] || 8 || +0%
|}

== Quality probability ==
When an item gets produced and the initial roll decides that the quality of the item will increase, there is a 90% chance it will rise one tier, a 9% chance it will rise two, a 0.9% chance it will rise three, and a 0.1% chance it will rise four. This is of course capped if the item already started out at a higher tier.

== Mathematical model ==
The mathematical model is time discrete. As opposed to dealing with derivatives in respect to time, the next state is a direct function of the previous state.

* <math>m_{i,k}</math> ... Number of materials of tier <math>i</math> (whereas 1 is common, 2 is uncommon, 3 is rare, 4 is epic, and 5 is legendary) after the <math>k</math>-th iteration before being crafted. (this doesn't mean that an item only needs one type of ingredient, but that "1 materials" can be crafted into 1 item from them)
* <math>n_{i,k}</math> ... Number of items of tier <math>i</math> after the <math>k</math>-th iteration, after being crafted together.
* <math>p_0</math> ... the crafting machines inherent productivity bonus
* <math>N</math> ... number of modules the crafting machine can hold
* <math>q_r = 4 \cdot 6.2 \, \% = 0.248</math> ... quality probability of the recyclers with 4 [[quality module 3]]'s (6.2% is the chance of a legendary tier quality module 3)
* <math>p_i = p_0 + x_i \cdot 25 \, \%</math> ... productivity due to <math>x_i</math> legendary [[productivity module 3]]'s in the crafting machine which takes <math>i</math>-tier materials (25% is the productivity boost of a legendary tier productivity module 3)
* <math>q_i = (N-x_i) \cdot 6.2 \, \%</math> ... quality probability due to <math>(N - x_i)</math> legendary quality module 3's in the crafting machine which takes <math>i</math>-tier materials

=== Recycled materials ===
When a quarter of all items being recycled and the quality probability rules, we can write the equations for the amount of materials received after recycling. This calculation is rather simple, as uncommon items can only come forth from uncommon material failing to raise in quality. The second tier are 90% of tier 1 items that did increase, and all those tier 2 items that didn't. This continues for all tiers as follows, but legendary items <math>n_{5,k}</math> will not be recycled

<math>
\begin{align}
\text{common material:}\quad m_{1,k+1} &= \frac{n_{1,k}}{4} \cdot (1 - q_r)\\
\text{uncommon material:}\quad m_{2,k+1} &= \frac{n_{1,k}}{4} \cdot q_r \cdot 0.9 &+& \frac{n_{2,k}}{4} \cdot (1 - q_r)\\
\text{rare material:}\quad m_{3,k+1} &= \frac{n_{1,k}}{4} \cdot q_r \cdot 0.09 &+& \frac{n_{2,k}}{4} \cdot q_r \cdot 0.9 &+& \frac{n_{3,k}}{4} \cdot (1 - q_r)\\
\text{epic material:}\quad m_{4,k+1} &= \frac{n_{1,k}}{4} \cdot q_r \cdot 0.009 &+& \frac{n_{2,k}}{4} \cdot q_r \cdot 0.09 &+& \frac{n_{3,k}}{4} \cdot q_r \cdot 0.9 &+& \frac{n_{4,k}}{4} \cdot (1 - q_r)\\
\text{legendary material:}\quad m_{5,k+1} &= \frac{n_{1,k}}{4} \cdot q_r \cdot 0.001 &+& \frac{n_{2,k}}{4} \cdot q_r \cdot 0.01 &+& \frac{n_{3,k}}{4} \cdot q_r \cdot 0.1 &+& \frac{n_{4,k}}{4} \cdot q_r\\
\end{align}
</math>

which can also be written as a vector-matrix-multiplication <math>\textbf{m}_{k+1} = \textbf{A} \,\textbf{n}_k</math>

<math>
\underbrace{\begin{bmatrix} m_{1,k+1}\\ m_{2,k+1}\\ m_{3,k+1}\\ m_{4,k+1}\\ m_{5,k+1}\\ \end{bmatrix}}_{\textbf{m}_{k+1}}
=
\underbrace{\begin{bmatrix}
\frac{1 - q_r}{4} & 0 & 0 & 0 & 0\\
\frac{0.9 \, q_r}{4} & \frac{1 - q_r}{4} & 0 & 0 & 0\\
\frac{0.09 \, q_r}{4} & \frac{0.9 \, q_r}{4} & \frac{1 - q_r}{4} & 0 & 0 \\
\frac{0.009 \, q_r}{4} & \frac{0.09 \, q_r}{4} & \frac{0.9 \, q_r}{4} & \frac{1 - q_r}{4} & 0\\
\frac{0.001 \, q_r}{4} & \frac{0.01 \, q_r}{4} & \frac{0.1 \, q_r}{4} & \frac{q_r}{4} & 0\\
\end{bmatrix}}_{\textbf{A}}
\underbrace{\begin{bmatrix} n_{1,k}\\ n_{2,k}\\ n_{3,k}\\ n_{4,k}\\ n_{5,k}\\ \end{bmatrix}}_{\textbf{n}_k}
</math>

As we assume there is nothing to be done about recyclers to make them more effective but fill them with all quality modules, the value of <math>q_r = 0.248</math> is a constant, and therefore the matrix <math>\textbf{A}</math> is also a constant. For legendary quality module 3's, it looks like:

<math> \textbf{A} =
\begin{bmatrix}
0.188 & 0 & 0 & 0 & 0 \\
0.055\,8 & 0.188 & 0 & 0 & 0 \\
0.005\,58 & 0.055\,8 & 0.188 & 0 & 0 \\
0.000\,558 & 0.005\,58 & 0.055\,8 & 0.188 & 0 \\
0.000\,062 & 0.000\,62 & 0.006\,2 & 0.062 & 0 \\
\end{bmatrix}</math>

=== Recrafted items ===
Once more we first check how many items are produced, which is, again, the sum of all possible ways to get to a tier, this time adding productivity <math>p_i</math>, and bringing along all those items which are already legendary <math>n_{5,k}</math>

<math>
\begin{align}
\text{common item:}\quad n_{1,k+1} &= m_{1,k+1} \cdot ( 1 + p_1 ) \cdot (1 - q_1)\\
\text{uncommon items:}\quad n_{2,k+1} &= m_{1,k+1} \cdot ( 1 + p_1 ) \cdot q_1 \cdot 0.9 &+& m_{2,k+1} \cdot ( 1 + p_2 ) \cdot (1 - q_2)\\
\text{rare items:}\quad n_{3,k+1} &= m_{1,k+1} \cdot ( 1 + p_1 ) \cdot q_1 \cdot 0.09 &+& m_{2,k+1} \cdot ( 1 + p_2 ) \cdot q_2 \cdot 0.9 &+& m_{3,k+1} \cdot ( 1 + p_3 ) \cdot (1 - q_3) \\
\text{epic items:}\quad n_{4,k+1} &= m_{1,k+1} \cdot ( 1 + p_1 ) \cdot q_1 \cdot 0.009 &+& m_{2,k+1} \cdot ( 1 + p_2 ) \cdot q_2 \cdot 0.09 &+& m_{3,k+1} \cdot ( 1 + p_3 ) \cdot q_3 \cdot 0.9 &+& m_{4,k+1} \cdot ( 1 + p_4) \cdot (1 - q_4) \\
\text{legendary items:}\quad n_{5,k+1} &= m_{1,k+1} \cdot ( 1 + p_1 ) \cdot q_1 \cdot 0.001 &+& m_{2,k+1} \cdot ( 1 + p_2 ) \cdot q_2 \cdot 0.01 &+& m_{3,k+1} \cdot ( 1 + p_3 ) \cdot q_3 \cdot 0.1 &+& m_{4,k+1} \cdot ( 1 + p_4) \cdot q_4 &+& m_{5,k+1} \cdot ( 1 + p_5) + n_{5,k}\\
\end{align}
</math>

This can once more be written as a vector-matrix-multiplication <math>\textbf{n}_{k+1} = \textbf{B} \,\textbf{m}_{k+1} + \textbf{C} \,\textbf{n}_{k}</math>

<math>
\underbrace{\begin{bmatrix} n_{1,k+1}\\ n_{2,k+1}\\ n_{3,k+1}\\ n_{4,k+1}\\ n_{5,k+1}\\ \end{bmatrix}}_{\textbf{n}_{k+1}} =
\underbrace{\begin{bmatrix}
( 1 + p_1 ) \cdot (1 - q_1) & 0 & 0 & 0 & 0 \\
( 1 + p_1 ) \cdot q_1 \cdot 0.9 & ( 1 + p_2 ) \cdot (1 - q_2) & 0 & 0 & 0 \\
( 1 + p_1 ) \cdot q_1 \cdot 0.09 & ( 1 + p_2 ) \cdot q_2 \cdot 0.9 & ( 1 + p_3 ) \cdot (1 - q_3) & 0 & 0 \\
( 1 + p_1 ) \cdot q_1 \cdot 0.009 & ( 1 + p_2 ) \cdot q_2 \cdot 0.09 & ( 1 + p_3 ) \cdot q_3 \cdot 0.9 & ( 1 + p_4) \cdot (1 - q_4) & 0 \\
( 1 + p_1 ) \cdot q_1 \cdot 0.001 & ( 1 + p_2 ) \cdot q_2 \cdot 0.01 & ( 1 + p_3 ) \cdot q_3 \cdot 0.1 & ( 1 + p_4) \cdot q_4 & ( 1 + p_5)\\
\end{bmatrix}}_{B}
\underbrace{\begin{bmatrix} m_{1,k+1}\\ m_{2,k+1}\\ m_{3,k+1}\\ m_{4,k+1}\\ m_{5,k+1}\\ \end{bmatrix}}_{\textbf{m}_{k+1}}
+
\underbrace{\begin{bmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&0&0\\
0&0&0&0\\
0&0&0&1\\
\end{bmatrix}}_{\textbf{C}}
\underbrace{\begin{bmatrix} n_{1,k}\\ n_{2,k}\\ n_{3,k}\\ n_{4,k}\\ n_{5,k}\\ \end{bmatrix}}_{\textbf{n}_k}
</math>

=== Combined model ===

As such we gain an equation for the amount of items for every tier after any amount of iterations

<math>
\begin{align}
\textbf{n}_{k+1}
&= \textbf{B} \, \textbf{m}_{k+1} + \textbf{C} \, \textbf{n}_{k}\\
&= \textbf{B} \, \textbf{A} \,\textbf{n}_{k} + \textbf{C} \, \textbf{n}_{k}\\
&= (\textbf{B} \, \textbf{A} +\textbf{C} ) \, \textbf{n}_{k}\\
&= \textbf{M} \, \textbf{n}_{k}\\
\textbf{n}_{k+2} &= \textbf{Q} \, \textbf{n}_{k+1} = \textbf{Q} \, \textbf{Q} \, \textbf{n}_{k} = \textbf{Q}^2 \, \textbf{n}_{k}\\
&\vdots\\
\textbf{n}_k &= \textbf{Q}^k \, \textbf{n}_0\\
\end{align}
</math>

A realistic starting point for <math>\textbf{n}_0</math> is only items in tier 1, and if we do not care for any other tiers, we simply choose sufficiently large <math>k = 100</math>, and, through numeric means, try to maximize a single value. <math>q_{100,\text{crit}}</math>, which falls in the fifth row, first column, of the Matrix <math>\textbf{Q}</math> raised to the 100th power.

<math>
\begin{align}
\textbf{n}_k &= Q^k \, \textbf{n}_0\\
\begin{bmatrix}*\\ *\\ *\\ *\\ n_{5,100} \end{bmatrix} &=
\begin{bmatrix}
* & * & * & * \\
* & * & * & * \\
* & * & * & * \\
* & * & * & * \\
q_{100,\text{crit}} & * & * & * \\
\end{bmatrix} \begin{bmatrix}n_{1,0}\\ *\\ *\\ *\\ * \end{bmatrix}\\
\end{align}
</math>

The choice of optimization method (simplex, branch and bound, etc.) itself is irrelevant, although most software will want the problem to be stated in a way so it can find a minimum, and may require the proper guard rails as to not pick values below 0 or higher than the maximum number of allowed modules.